Circle A has a radius of #2 # and a center at #(3 ,6 )#. Circle B has a radius of #5 # and a center at #(2 ,3 )#. If circle B is translated by #<-2 ,1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
1 Answer
circles overlap.
Explanation:
What we have to do here is compare the distance (d) between the centres of the circles to the sum of the radii.
• If sum of radii > d , then circles overlap
• If sum of radii < d , then no overlap
The first step is to find the new centre of B under the translation. A translation does not change the shape of a figure , only it's position.
Under a translation of
#((-2),(1))# centre B(2 ,3) → (2-2 ,3+1) → B(0 ,4)-(new centre)
To calculate the distance (d) between the centres use the
#color(blue)" distance formula"#
#color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))#
where# (x_1,y_1)" and " (x_2,y_2)" are 2 points"# The 2 points here being the centres of A and B.
let
# (x_1,y_1)=(3,6)" and " (x_2,y_2)=(0,4)#
#d=sqrt((0-3)^2+(4-6)^2)=sqrt(9+4)=sqrt13≈3.606# radius of A + radius of B = 2 + 5 = 7
Since sum of radii > d , then circles overlap.
